https://doi.org/10.14317/jami.2017.045
BIVARIATE DYNAMIC CUMULATIVE RESIDUAL TSALLIS ENTROPY†
MADAN MOHAN SATI∗AND HARINDER SINGH
Abstract. Recently, Sati and Gupta (2015) proposed two measures of uncertainty based on non-extensive entropy, called the dynamic cumulative residual Tsallis entropy (DCRTE) and the empirical cumulative Tsallis entropy. In the present paper, we extend the definition of DCRTE into the bivariate setup and study its properties in the context of reliability theory.
We also define a new class of life distributions based on bivariate DCRTE.
AMS Mathematics Subject Classification : 65H05, 65F10.
Key words and phrases : Tsallis entropy, Cumulative residual entropy, Bivariate Hazard rate and Bivariate mean residual life function.
1. Introduction
The concept of entropy was introduced by Shannon [21]. Let X be a continu- ous random variable having probability density function f (x), cumulative distri- bution function F (x), survival function ¯F(x) and hazard rate r(x) = f (x)/ ¯F(x).
Then the differential form of Shannon entropy is defined as H(X) =−
∫ ∞
0
f(x) log (f (x)) dx. (1)
The entropy measure (1) is not useful for a system which has survived up to age t, Ebrahimi [5] modified (1) to measure uncertainty for the residual life time Xt= [X− t|X > t], where t > 0 given by
H(X; t) =−
∫∞
t
f(x) F¯(t) log
(f(x) F(t)¯ )
dx. (2)
Received November 20, 2015. Revised November 10, 2016. Accepted November 15, 2016.
∗Corresponding author. †This work was supported by the research grant of the University
⃝ 2017 Korean SIGCAM and KSCAMc . 45
Rao et al. [16] introduced a new measure of uncertainty based on survival func- tion instead of probability density function of a random variable X as follows:
ξ(X) =−
∫ ∞
0
F(x) log¯ (F(x)¯ )
dx, (3)
and called it the Cumulative Residual Entropy (CRE).
Asadi and Zohrevand [2] proposed a new measure of uncertainty for residual lifetime Xt= [X− t|X > t], which is given as
ξ(X; t) =−
∫∞
t
F¯(x) F¯(t) log
(F¯(x) F¯(t)
)
dx, (4)
and called it Dynamic Cumulative Residual Entropy (DCRE).
The properties and applications of DCRE have been studied extensively by Asadi and Zohervand [2], Di Crescenzo and Longobardi [4], Abbasnejad et al. [1], Navarro et al. [11], Sunoj and Linu [22], Kumar and Taneja [9], Sati and Gupta [20].
In the literature several generalization of Shannon entropy are available such as Renyi entropy [17], Varma entropy [24], Tsallis entropy [23] etc. In this article, we focus on non-extensive entropy.
Tsallis [23] defined the generalized non-expansive entropy of order α as Sα(X) = 1
α− 1 (
1−
∫ ∞
0
(f (x))αdx )
, α >0, α̸= 1. (5) In a recent work, Sati and Gupta [20] have proposed Dynamic Cumulative Resid- ual Tsallis Entropy (DCRTE) of order α as follows:
ηα(X; t) = 1 α− 1
1 −
∫∞
t
(F(x)¯ F¯(t)
)α
dx
, α > 0 , α ̸= 1, (6) and studied its properties and applications.
The multivariate life distributions are used for studying the reliability char- acteristics of multi-component system with each component having a lifetime depending on the next component. In the univariate case, the reliability char- acteristics can be extended to higher dimensions. Although, a lot of work has been done on information measures in the univariate case, but very limited work has been done in higher dimensions. For more details, we refer to Rajesh and Nair [12], Nadarajah and Zografos [10], Ebrahimi et al. [6], Sathar et al. [19]
and Rajesh et al. [13], [14], [15].
The main objective of the paper is to extend DCRTE defined in (6) to bivari- ate setup and study its properties and connect it to some well known reliability models. In section 2, we propose a bivariate dynamic cumulative residual Tsallis entropy (BDCRTE) of order α and characterize some well known bivariate mod- els using the BDCRTE. In section 3, we define new classes of life distributions based on BDCRTE and study their properties.
2. Bivariate Dynamic Cumulative Residual Tsallis Entropy(BDCRTE)
In this section, we extend the definition of DCRTE to the bivariate setup known as the bivariate cumulative residual Tsallis entropy (BDCRTE) and we also give some characterization results of well known bivariate distributions in term of BDCRTE.
Definition 2.1. Let X = (X1, X2) be a bivariate random vector admitting an absolutely continuous probability density function f (x1, x2), cumulative density function F (x1, x2) and survival function ¯F(x1, x2) with respect to Lebesgue measure in the positive octant R+2 ={(t1, t2)|ti>0, i = 1, 2} of the two-dimensional Euclidean space R2. We define the bivariate DCRTE as
ηα(X; t1, t2) = 1 α− 1
1 −
∫∞
t1
∫∞
t2
(F(x¯ 1, x2) F¯(t1, t2)
)α
dx2dx1
, α > 0 , α ̸= 1.
(7) Ebrahimi [6] has proved that the bivariate residual entropy is not invariant un- der non singular transformations. Similarly we can show that bivariate DCRTE defined in equation (7) is not invariant under non singular transformations.
If Yj = ϕj(Xj), j = 1, 2 are one to one transformations, then
ηα(Yj; ϕ1(t1), ϕ2(t2)) = 1 α− 1
1 −
∫∞
t1
∫∞
t2
(F¯(x1, x2) F¯(t1, t2)
)α
J dx2dx1
,
where J =| ∂
∂x1ϕ1(x1)× ∂
∂x2ϕ2(x2)| is the absolute value of the Jacobian of transformation. In particular, if we take ϕj(Xj) = ajXj+ bj, then we get
ηα(Yj; ϕ1(t1), ϕ2(t2)) = (1− a1a2)
(α− 1) + a1a2ηα(X; t1, t2).
Now we take into account the behavior of the dynamic cumulative residual Tsallis entropy for the conditional distributions. Let us consider the random variables Yj = (Xj|Xi > ti, i, j = 1, 2; i̸= j), where Yj, j = 1, 2 corresponds to the conditional distributions of Xj given that Xi has survived up to time ti, i = 1, 2 and have the survival functions F¯(x1, t2)
F(t¯ 1, t2) for x1 ≥ t1 & F(t¯ 1, x2) F¯(t1, t2) for x2≥ t2, respectively. The DCRTE for the random variables Yj, j = 1, 2 are defined as follows:
η1α(X; t1, t2) = 1 α− 1
1 −
∫∞
t1
(F¯(x1, t2) F(t¯ 1, t2)
)α
dx1
, α > 0 , α ̸= 1, (8)
and
η2α(X; t1, t2) = 1 α− 1
1 −
∫∞
t2
(F¯(t1, x2) F¯(t1, t2)
)α
dx2
, α > 0 , α ̸= 1, (9) respectively.
For a bivariate random vector X = (X1, X2), Johnson and Kotz [8] defined the bivariate hazard rate as
r(t1, t2) = ( r1(t1, t2), r2(t1, t2) ), where
ri(t1, t2) =− ∂
∂ti
log ¯F(t1, t2), i = 1, 2. (10) For a bivariate random vector X = (X1, X2), Zahedi [25] defined the bivariate mean residual life function (MRLF) as
m(t1, t2) = ( m1(t1, t2), m2(t1, t2) ), where
mi(t1, t2) =(F¯(t1, t2))−1∫∞
ti
F(t¯ 1, t2) dti, i= 1, 2. (11)
The following theorem shows that the bivariate dynamic cumulative residual Tsallis entropy (BDCRTE) uniquely determines the survival function ¯F(t1, t2).
Theorem 2.2. Let X = (X1, X2) be a non-negative random vector admit- ting continuous distribution function with respect to Lebsegue measure. Let ηiα(X; t1, t2) < ∞ ; i = 1, 2, t = (t1, t2)≥ 0; ∀ α > 0 (̸= 1). Then for each α, ηiα(X; t1, t2) (where as ∂t∂
iηiα(X; t1, t2) ̸= 0, ∀ i = 1, 2 ) uniquely deter- mines the survival function ¯F(t1, t2).
Proof. From the equation (8), we have
(α− 1) η1α(X; t1, t2) = 1−
∫∞ t1
(F¯(x1, t2))α
dx1
(F(t¯ 1, t2))α . (12) Differentiating (12) with respect to t1and simplifying, we obtain
(α− 1) ∂
∂t1η1α(X; t1, t2) = 1 + α r1(X; t1, t2) [(α− 1)η1α(X; t1, t2)− 1]. (13) Similarly for i = 2, we also get
(α− 1) ∂
∂t2
η2α(X; t1, t2) = 1 + α r2(X; t1, t2) [(α− 1)η2α(X; t1, t2)− 1]. (14) Let ¯FX(t1, t2) and ¯FY(t1, t2) be two survival functions having bivariate dy- namic entropies ηiα(X; t1, t2) and ηiα(Y ; t1, t2) with hazard rates ri(X; t1t2) and ri(Y ; t1t2), i = 1, 2 respectively.
Consider the following relationship between entropies of random vector X and Y :
ηiα(X; t1, t2) = ηiα(Y ; t1, t2) , i = 1, 2. (15) Taking i = 1 and differentiating (15) with respect to t1, we get
∂
∂t1η1α(X; t1, t2) = ∂
∂t1η1α(Y ; t1, t2).
(α− 1) ∂
∂t1η1α(X; t1, t2) = (α− 1) ∂
∂t1η1α(Y ; t1, t2). (16) Using (13), the equation (16) becomes
1 + α r1(X; t1, t2) [(α− 1)η1α(X; t1, t2)− 1] = 1 + α r1(Y ; t1, t2) [(α− 1)η1α(Y ; t1, t2)− 1].
(17)
Since η1α(X; t1, t2) = η1α(Y ; t1, t2), therefore the equation (17) reduces to r1(X; t1t2) = r1(Y ; t1t2).
Similarly for i = 2, we get
r2(X; t1t2) = r2(Y ; t1t2).
Thus, we have
F¯X(t1, t2) = ¯FY(t1, t2).
Hence, ηiα(X; t1, t2) uniquely determines the survival function ¯F(t1, t2). The following theorem characterize some well known bivariate distributions using relationship between BDCRTE and bivariate mean residual life function m(X; t1, t2).
Theorem 2.3. For the random vector X = (X1, X2) admitting continuous distribution function with respect to Lebsegue measure, a relationship of the form (α− 1) ηiα(X; t1, t2) = 1− Kmi(X; t1, t2) , i = 1, 2, α > 0 , α̸= 1, (18) where mi(X; t1, t2) , i = 1, 2 are the components of the bivariate mean residual life function and K is a constant independent of ti, holds for all ti ≥ 0, if and only if X follows any one of the three distributions:
(i) the bivariate Pareto distribution with joint survival function F(t¯ 1, t2) = (1 + a1t1+ a2t2+ bt1t2)−c; a1, a2, c, t1, t2>0;
0 < b < (c + 1)a1a2, (19) (ii) the Gumbel’s bivariate exponential distribution with joint survival function
F¯(t1, t2) = exp (−λ1t1− λ2t2− θt1t2) ; λ1, λ2, t1, t2>0;
0 < θ < λ1λ2, (20) and
(iii) the bivariate finite range distribution with joint survival function F¯(t1, t2) = (1− p1t1− p2t2+ qt1t2)d; p1, p2, d >0; 0 < t1< 1
p1
; 0 < t2<1− p1t1
p2− qt1
, (21)
according as Kα <1, Kα = 1 and Kα > 1, respectively.
Proof. Differentiating equation (18) with respect to t1by taking i = 1, we obtain (α− 1) ∂
∂t1
η1α(X; t1, t2) = −K ∂
∂t1
m1(X; t1, t2).
Using the equation (13), we get
1− Kα r1(X; t1, t2) m1(X; t1, t2) = −K ∂
∂t1
m1(X; t1, t2). (22) Using the relation r1(X; t1, t2) m1(X; t1, t2) = 1 +∂t∂1m1(X; t1, t2), the equation (22) reduces to
∂
∂t1m1(X; t1, t2) =(Kα− 1) K(1− α) = C.
Integrating on both side with respect to t1, we get
m1(X; t1, t2) = Ct1+ D1(t2), (23) where D1 is independent of t1.
Similarly for i = 2, we have
m2(X; t1, t2) = Ct2+ D2(t1). (24) Hence
mi(X; t1, t2) = Cti+ Di(tj) , i̸= j, i, j = 1, 2,
where C = (KαK(1−α)−1) and Di(tj) is a function of tj only. Based on the character- ization theorem given by Sankaran and Nair [18], we can easily prove that X follows bivariate Pareto distribution with survival function (19) when C > 0, Gumbel’s exponential distribution with survival function (20) when C = 0 and bivariate finite range distribution with survival function (21) when C < 0.
Converse:
(i) When X follows bivariate Pareto distribution with survival function (19), then using the equation (8), we get
(α− 1) η1α(X; t1, t2) = 1−
∞∫
t1
(1 + a1x1+ a2t2+ bx1t2)−cα dx1
(1 + a1t1+ a2t2+ bt1t2)−cα
= 1−
[ (c− 1) (cα− 1)
(1 + a1t1+ a2t2+ bt1t2) (c− 1)(a1+ bt2)
] .
Similar result holds for i = 2. Hence
(α− 1) ηiα(X; t1, t2) = 1− Kmi(X; t1, t2) , where K = (cα(c−1)−1), such that Kα < 1.
(ii) When X follows Gumbel’s exponential distribution with survival function (20), then using the equation (8), we get
(α− 1) η1α(X; t1, t2) = 1−
∫∞ t1
(e−λ1x1−λ2t2−θx1t2)α
dx1 (e−λ1t1−λ2t2−θt1t2)α
= 1− 1
α(λ1+ θt2). Similar result holds for i = 2. Hence
(α− 1) ηiα(X; t1, t2) = 1− Kmi(X; t1, t2) , where K = α1, such that Kα = 1.
(iii) When X follows bivariate finite range distribution with survival function (21), then using the equation (8), we get
(α− 1) η1α(X; t1, t2) = 1−
∞∫
t1
(1− p1x1− p2t2+ qx1t2)dαdx1
(1− p1t1− p2t2+ qt1t2)dα
= 1−
[ (d + 1) (dα + 1)
(1− p1t1− p2t2+ qt1t2) (d + 1)(p1+ qt2)
] . Similar result holds for i = 2. Hence
(α− 1) ηiα(X; t1, t2) = 1− Kmi(X; t1, t2) , where K = (dα+1)(d+1) , such that Kα > 1.
Now we provide characterization result in terms of relationship between bi- variate DCRTE and bivariate hazard rate function.
Theorem 2.4. For the random vector X = (X1, X2) admitting an absolutely continuous function with respect to Lebesgue measure. Then the following rela- tionship of the form
(α− 1) ∂
∂tiηiα(X; t1, t2) = c ri(X; t1, t2) , i= 1, 2, α > 0, α̸= 1, (25) hold for all t1, t2≥ 0, then X follows the Gumbel’s bivariate exponential distri- bution with joint survival function
F(t¯ 1, t2) = exp (−λ1t1− λ2t2− θt1t2) ; λ1, λ2, t1, t2>0; 0 < θ < λ1λ2, when c= 0.
(26)
Proof. When equation (25) hold for i = 1, then using the equation (12) and equation (13), we get
1− α r1(X; t1, t2)
∫∞ t1
(F(x¯ 1, t2))α
dx1
(F¯(t1, t2))α = c r1(X; t1, t2), or equivalently
1
r1(X; t1, t2)= c + α
∫∞ t1
(F¯(x1, t2))α
dx1
(F¯(t1, t2))α . (27) Differentiating equation (27) with respect to t1and simplifying, we obtain
−1 (r1(X; t1, t2))2
∂
∂t1{r1(X; t1, t2)} = − αc r1(X; t1, t2).
∂
∂t1{log (r1(X; t1, t2))} = αc r12(X; t1, t2). (28) For simplification, we assume that log (r1(X; t1, t2)) = y1(t1, t2) ; that is, r1(X; t1, t2) = ey1(t1,t2), then the equation (28) reduces to
∂
∂t1{(y1(t1, t2))} = αc e2y1(t1,t2). Integrating on both side with respect to t1, we get
r1(X; t1, t2) =√ 1
K1(t2)− 2αc t1
. Similarly for i = 2, we get
r2(X; t1, t2) = 1
√K2(t1)− 2αc t2
. Hence
ri(X; t1, t2) =√ 1
Ki(tj)− 2αc ti
, i̸= j , i, j = 1, 2, (29) where Ki(tj) > 0 is constant and independent of ti.
When c = 0, then from the equation (29), we get ri(X; t1, t2) = √ 1
Ki(tj) or equivalently
− ∂
∂ti
{log ¯F(t1, t2)}
= 1
√Ki(tj). Integrating both side with respect to ti, we get
−log ¯F(t1, t2) = ti
√Ki(tj)+ Qi(tj)
F¯(t1, t2) = e−
[
√ ti
Ki(tj )+ Qi(tj) ]
, i̸= j , i = 1, 2. (30)
Applying for i = 1, 2 and equating (30) we get t1
√K1(t2)+ Q1(t2) = t2
√K2(t1)+ Q2(t1) (31)
As t1→ 0, equation (31) becomes Q1(t2) = t2
√K2(0)+ Q2(0)
As t2→ 0, equation (31) becomes Q2(t1) = t1
√K1(0)+ Q1(0)
Putting the value of Q1(t2) and Q2(t1) in the equation (31), we get t1
√K1(t2)+√ t2
K2(0)+ Q2(0) = √ t2
K2(t1)+√t1
K1(0)+ Q1(0) (32) Since Q1(0) = Q2(0) = ¯F(0, 0), equation (31) become
1 t2
√K1(t2)− 1 t2
√K1(0) = 1 t1
√K2(t1)− 1 t1
√K2(0) = θ(say), (33) which implies
√ 1
K1(t2)= λ1+ θt2
Similarly, we get
√ 1
K2(t1)= λ2+ θt1, where √ 1
K1(0) = λ1and √ 1
K2(0) = λ2. Substituting these value in the equation (31), after simplification we get
F(t¯ 1, t2) = exp (−λ1t1− λ2t2− θt1t2) .
The converse part is staightforward.
3. New class of life distributions
In this section, we define a new class of life distributions based on proposed bivariate dynamic cumulative residual Tsallis entropy (BDCRTE).
Definition 3.1. The distribution function F (t1, t2) is said to be increasing (de- creasing) in Bivariate DCRTE, denoted by IBDCRTE (DBDCRTE), if ηiα(X; t1, t2) is an increasing (decreasing) function of ti, i = 1, 2.
The following theorem gives the necessary and sufficient conditions for BD- CRTE to be increasing(decreasing) BDCRTE.
Theorem 3.2. The bivariate distribution function F(t1, t2) is increasing (de- creasing) BDCRTE if and only if for all t1, t2≥ 0.
ηiα(X; t1, t2) ≥ (≤) 1 (α− 1)
(
1 − 1
α ri(t1, t2) )
, i= 1, 2∀ α > 0 , α ̸= 1.
Proof. The proof of the theorem follows from equation (13) and (14). The following theorem provides lower bound of BDCRTE based on bivariate mean residual life function.
Theorem 3.3. Let X = (X1, X2) be a non-negative random vector admitting absolute continuous distribution function with respect to Lebesgue measure and mi(t1, t2), i = 1, 2 are the components of the bivariate mean residual life function, then
ηiα(X; t1, t2) ≥ 1
(α− 1) (1 − mi(t1, t2) ) , i = 1, 2 ,∀ α > 0 , α ̸= 1.
Proof. We know that (F¯(t1, t2))α
≤ (≥) ¯F(t1, t2), ∀ ti>0, i = 1, 2 , α > 1 (0 < α < 1).
∫∞
t1
(F¯(x1, t2) F(t¯ 1, t2)
)α
dx1 ≤ (≥)
∫∞
t1
(F¯(x1, t2) F(t¯ 1, t2) )
dx1, α >1 (0 < α < 1).
Case 1: When α > 1 1
(α− 1)
1 −
∫∞
t1
(F¯(x1, t2) F¯(t1, t2)
)α
dx1
≥ 1
(α− 1)
1 −
∫∞
t1
(F¯(x1, t2) F¯(t1, t2) )
dx1
.
η1α(X; t1, t2) ≥ 1
(α− 1) (1 − m1(t1, t2) ) . Case 2: When 0 < α < 1
1 (1− α)
∫∞
t1
(F¯(x1, t2) F(t¯ 1, t2)
)α
dx1− 1
≥ 1
(1− α)
∫∞
t1
(F(x¯ 1, t2) F¯(t1, t2)
)
dx1− 1
.
η1α(X; t1, t2) ≥ 1
(α− 1) (1 − m1(t1, t2) ) . Thus
η1α(X; t1, t2) ≥ 1
(α− 1) (1 − m1(t1, t2) ) ,∀ α > 0 , α ̸= 1.
Similar result holds for i = 2. Therefore, we have ηiα(X; t1, t2) ≥ 1
(α− 1) (1 − mi(t1, t2) ) , i = 1, 2 ,∀ α > 0 , α ̸= 1.
We proposed the following theorem to obtain the bivariate hazard rate order- ing based on BDCRTE.
Theorem 3.4. Let X= (X1, X2) and Y = (Y1, Y2) be two non-negative random vector with survival functions ¯F(t1, t2) and ¯G(t1, t2), and hazard rates rF(t1, t2) and rG(t1, t2) respectively. If X≥hrY , that is rF(t1, t2)≤ rG(t1, t2), then
ηiα(X; t1, t2) ≤ (≥) ηiα(Y ; t1, t2) ,∀ α > 1 (0 < α < 1), i = 1, 2.
Proof. We know that rF(t1, t2)≤ rG(t1, t2) which implies ¯F(t1, t2)≥ ¯G(t1, t2).
Therefore, we have for all α > 0
∫∞
t1
(F¯(x1, t2) F¯(t1, t2)
)α
dx1 ≥
∫∞
t1
(G(x¯ 1, t2) G(t¯ 1, t2)
)α
dx1.
1−
∫∞
t1
(F¯(x1, t2) F¯(t1, t2)
)α
dx1 ≤ 1 −
∫∞
t1
(G(x¯ 1, t2) G¯(t1, t2)
)α
dx1.
For α > 1 1 (α− 1)
1 −
∫∞
t1
(F(x¯ 1, t2) F¯(t1, t2)
)α
dx1
≤ 1
(α− 1)
1 −
∫∞
t1
(G(x¯ 1, t2) G(t¯ 1, t2)
)α
dx1
η1α(X; t1, t2) ≤ η1α(Y ; t1, t2).
For 0 < α < 1 1
(1− α)
∫∞
t1
(F¯(x1, t2) F¯(t1, t2)
)α
dx1− 1
≥ 1
(1− α)
∫∞
t1
(G¯(x1, t2) G(t¯ 1, t2)
)α
dx1− 1
η1α(X; t1, t2) ≥ η1α(Y ; t1, t2).
Thus,
η1α(X; t1, t2) ≤ (≥) η1α(Y ; t1, t2) ,∀ α > 1(0 < α < 1) Similar result holds for i = 2. Therefore, we have
ηiα(X; t1, t2) ≤ (≥) ηiα(Y ; t1, t2) , ∀ α > 1(0 < α < 1), i = 1, 2.
Gupta and Sankaran [7] proposed the bivariate equilibrium distribution. Let X= (X1, X2) be a bivariate positive random vector admitting an absolute con- tinuous survival function ¯F(x1, x2). Then its bivariate equilibrium distribution is the distribution of a random vector Y = (Y1, Y2) such that the density function and survival function of (Yi| Yj> tj), i, j = 1, 2.i̸= j are of the form:
gi(ti|Yj> tj) = P(Xi> ti|Xj> tj) E(Xi|Xj> tj)
= F¯(t1, t2)
F¯j(tj) E(Xi|Xj> tj), i̸= j; i, j = 1, 2 (34) and
G¯i(ti|Yj > tj) =
∫∞
ti
gi(u|Yj > tj)du
= F¯(t1, t2) mi(t1, t2)
F¯j(tj) E(Xi|Xj> tj), i̸= j; i, j = 1, 2, (35) respectively, for t1, t2≥ 0.
Remark 3.1. The residual Tsallis entropy can be expressed for the bivariate random vector X = (X1, X2) as follows:
Hα(X; t1, t2) = ( H1α(X; t1, t2) , H2α(X; t1, t2) ) , where
Hiα(X; t1, t2) = 1 (α− 1)
1 −
∫∞
ti
(fi(xi|Xj> tj) F¯i(ti|Xj > tj)
)α
dxi
, i, j = 1, 2, i ̸= j . In the following theorem we establish a relation between BDCRTE and resid- ual Tsallis entropy corresponding to the bivariate equilibrium random vector Y = (Y1, Y2).
Theorem 3.5. Let X = (X1, X2) be a non-negative random vector and Y = (Y1, Y2) be the equilibrium random vector associate with X, then
Hiα(Y ; t1, t2) = ηiα(X; t1, t2)
mαi(t1, t2) +1− m−αi (t1, t2)
(α− 1) , i= 1, 2, ∀ α > 0 , α ̸= 1, (36) where Hiα(Y ; t1, t2) denote the bivariate residual Tsallis entropy corresponding to Y and mi(t1, t2), i = 1, 2 are the components of the bivariate mean residual life function.
Proof. When the equation (36) holds for i = 1, we have
H1α(Y ; t1, t2) = 1 (α− 1)
1 −
∫∞
t1
(g1(y1|Y2> t2) G¯1(t1|Y2> t2)
)α
dy1
.
Applying the results of the equations (34) and (35), we get H1α(Y ; t1, t2) = η1α(X; t1, t2)
mα1(t1, t2) +1− m−α1 (t1, t2) (α− 1) .
Similar result holds for i = 2. Therefore, we have Hiα(Y ; t1, t2) = ηiα(X; t1, t2)
mαi(t1, t2) +1− m−αi (t1, t2)
(α− 1) , i= 1, 2, ∀ α > 0 , α ̸= 1.
Cox (1972) introduced the concept of proportional hazards model (PHM). Let X and Xθ be two continuous random variables with survival functions ¯FX(x) and ¯FXθ(x), respectively. The relation between survival functions of random life times is given by
F¯Xθ(x) = [F¯X(x)]θ
, x ∈ R , θ > 0.
The following lemma compares the DCRTE of X and Xθ. Lemma 3.6. The following statement holds:
(a) ηα(Xθ; t) ≥ ηα(X; t) for θ≥ 1 , α > 1 and θ ≤ 1 , 0 < α < 1.
(b) ηα(Xθ; t) ≤ ηα(X; t) for θ≥ 1 , 0 < α < 1 and θ ≤ 1 , α > 1.
The proof of the lemma is straightforward.
4. Conclusion
The definition of dynamic cumulative residual Tsallis entropy (DCRTE) have been extended to bivariate setup consequently proposed the Bivariate DCRTE.
The monotonic behaviour in the context of bivariate random vector has also been studied. Some well known bivariate life time distributions are characterized.
Additionally, we have defined some a new class of life distributions based on BDCRTE.
References
1. M. Abbasnejad, N.R. Arghami, S. Morgenthaler and G.R. Mohtashami Borzadaran, On the dynamic survival entropyStatistics and Probability Letters 80 (2010) 1962-1971.
2. M. Asadi and Y. Zohrevand, On the dynamic cumulative residual entropy Journal of Statistical Planning and Inference 137 (2007) 1931-1941.
3. D.R. Cox and J. Roy, Regression models and life-tables (with discussion) Stat. Soc., Ser.
B 34 (1972) 187-220
4. A. Di Crescenzo and M. Longobardi, On cumulative entropies Journal of Statistical Plan- ning and Inference 139(12) (2009) 4072-4087.
5. N. Ebrahimi, How to measure uncertainty in the residual life time distribution Sankhya Ser. A 58 (1996) 48-56.
6. N. Ebrahimi, S.N.U.A. Kirmani and E.S. Soofi, Multivariate dynamic information J. Mul- tivariate Anal. 98 (2007) 328-349.
7. R.C. Gupta and P.G. Sankaran, Bivariate equilibrium distribution and its application to reliabilityCommunication in Statistic-Theory and Methods 27 (1998) 385-394.
8. N.L. Johnson and S. Kotz, A vector multivariate hazard rate Journal of Multivariate Anal- ysis 5 (1975) 53-66.
9. V. Kumar and H. Taneja, Some characterization results on generalized cumulative residual entropy measuresStatistics and Probability Letters 81 (2011) 1072-1077.
10. S. Nadarajah and K. Zografos, Expressions for Renyi and Shannon entropies for bivariate distributionsInform. Sci. 170 (2005) 173-189.
11. J. Navarro, Y. del Aguila and M. Asadi, Some new results on the cumulative residual entropyJournal of Statistical Planning and Inference 140 (2010) 310-322.
12. G. Rajesh and K.R. Nair, Residual entropy of conditional distribution Stat. Methods 2 (2000) 72-80.
13. G. Rajesh, A.E.I. Sathar and K.R. Nair, Bivariate extension of residual entropy and some characterization resultsJournal of the Indian Statistical Association 47 (2009) 91-107.
14. G. Rajesh, A.E.I. Sathar, K.R. Nair and K.V. Reshmi, Bivariate extension of dynamic cumulative residual entropyStatistical Methodology 16 (2014a) 72-82.
15. G. Rajesh, A.E.I. Sathar, K.V. Reshmi and K.R. Nair, Bivariate generalized cumulative residual entropySankhya A: The Indian Journal of Statistics 76 (2014b) 101-122.
16. M. Rao, Y. Chen, B.C. Vemuri and F. Wang, Cumulative residual entropy: a new measure of informationIEEE Transactions on Information Theory 50 (2004) 1220-1228.
17. A. Renyi, On measures of entropy and information, In: Neyman, J. (Ed.), Proceedings of the Fourth Berkeley Symposium on Mathematics, Statistics and Probability, Vol. I.
University of California Press, Berkeley, CA, (1961) 547-561.
18. P.G. Sankaran and N.U. Nair, Conditional mean residual life functions Communications in Statistics-Theory and Methods 29 (2000) 1663-1675.
19. A.E.I. Sathar, K.R.M. Nair and G. Rajesh, Generalized bivariate residual entropy function and some characterization resultsSouth African Statist. J. 44 (2009) 1-18.
20. M.M. Sati and N. Gupta, Some characterization results on dynamic cumulative residual Tsallis entropyJournal of Probability and Statistics (2015).
21. C.E. Shannon, A mathematical theory of communication The Bell Syst. Tech. Journal 27 (1948) 379-423.
22. S.M. Sunoj and M.N. Linu, Dynamic cumulative residual Renyis entropy A Journal of Theoretical and Applied Statistics 46 (2010) 41-56.
23. C. Tsallis, Possible Generalizations of Boltzmann-Gibbs Statistics Journal of Statistical Physics 52 (1998) 479-487.
24. R.S. Varma, Generalization of Renyi’s entropy of order α Journal of Mathematical Sci- ences 1 (1966) 34-48.
25. H. Zahedi, Some new classes of multivariate survival distribution functions Journal of Statistical Planning and Inference 11 (1985) 171-188.
Madan Mohan Sati received M.Sc. from Kumaun University Nainital, Uttrakhand, India.
And pursuing his Ph.D. in Mathematics from Jaypee University of Information Technology Waknaghat, Solan, India. His area of interests Information measures and reliability theory.
Department of Mathematics, Jaypee University of Information Technology Waknaghat, (H.P.), India.
e-mail: [email protected]
Harinder Singhis working as an Professor and Head Department of Mathematics, Jaypee University of Information Technology Waknaghat, Solan, (H.P.), Pin-173234, India.
Department of Mathematics, Jaypee University of Information Technology Waknaghat, Solan, (H.P.), Pin-173234, India.
e-mail: [email protected]
